= Poisson kernel of a three-dimensional quarter-space
{c}
{title2=$P=\frac z{2\pi}(r_-^{-3}-r_+^{-3})$}
For the face $z=0$ of $x>0,z>0$, set $r_\mp^2=(x\mp\xi)^2+(y-\eta)^2+z^2$, with source boundary coordinates $\xi>0$, $\eta\in\mathbb R$. The <Poisson kernel> for data on that face and zero data on $x=0$ is the displayed positive difference. It follows by differentiating the <Dirichlet Green function for a three-dimensional quarter-space> in the inward source direction. Integrating it against bounded continuous face data gives the bounded harmonic solution away from compatibility issues at the edge. Without a boundedness or suitable infinity condition, uniqueness fails because the <harmonic function> $xz$ vanishes on both faces.
Its integral over this one face is $2\arctan(x/z)/\pi$, the <harmonic measure> of that face. To calculate it, integration in $\eta$ first gives $z\{[(x-\xi)^2+z^2]^{-1}-[(x+\xi)^2+z^2]^{-1}\}/\pi$; the remaining half-line integral is an elementary arctangent difference. This mass lies between zero and one: the other face carries the remaining boundary influence.
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